Invariant Measures on Polarized Submanifolds in Group Quantization

TL;DR

This paper constructs explicit quasi-invariant measures on polarized coadjoint orbits of Lie groups, restoring measure invariance and unitarity in group quantization, exemplified by representations of SL(2,R).

Contribution

It introduces a method using central extensions to achieve invariant measures on coadjoint orbits, enabling unitarity in quantization.

Findings

Explicit measures are constructed for polarized coadjoint orbits.

Restoration of measure invariance leads to unitarity in quantization.

Representations of SL(2,R) are successfully recovered.

Abstract

We provide an explicit construction of quasi-invariant measures on polarized coadjoint orbits of a Lie group G. The use of specific (trivial) central extensions of G by the multiplicative group ${R}^+$ allows us to restore the strict invariance of the measures and, accordingly, the unitarity of the quantization of coadjoint orbits. As an example, the representations of $SL(2,\Real)$ are recovered.